The Republican Party's steep deficit among Latino voters last November will boost a new effort by the Obama administration to overhaul U.S. immigration laws, leading lawmakers from both parties said Sunday.

President Barack Obama is expected to kick off the push Tuesday with a speech in Las Vegas and to lay out more specifics during his State of the Union address on February 12. Obama has called addressing immigration the top legislative priority of his second term, and a group of Democratic and Republican senators says it plans to lay out its framework for a bill this week.

"First of all, Americans support it in poll after poll," one of those lawmakers, Sen. Robert Menendez, D-New Jersey, told ABC's "This Week."

Though many activists were unhappy with Obama for failing to tackle the issue in his first term, he won 71% of the Latino vote in November against Republican challenger Mitt Romney -- who had said he would drive undocumented workers to "self-deportation" by making conditions so harsh for them that they left the United States.

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Carlos Gutierrez, who was secretary of commerce during President George W. Bush's second term, told CNN's "State of the Union" that while immigration isn't the top issue for Latinos, "what they sense is that we don't welcome them. And we have to be the party that celebrates immigration."

Sen. John McCain, a veteran of failed attempts to address the issue during the Bush administration, said the proposals he and the other senators are hammering out aren't "that much different from what we tried to do in 2007." And the Arizona Republican told ABC they will include offering a path to citizenship for the estimated 11 million undocumented immigrants in the United States -- the issue that effectively sank previous bills.

This time, "There is a new, I think, appreciation on both sides of the aisle -- including maybe more importantly on the Republican side of the aisle -- that we have to enact a comprehensive immigration reform bill," said McCain, the party's 2008 presidential nominee.

"We are losing dramatically the Hispanic vote, which we think should be ours, for a variety of reasons, and we've got to understand that," McCain added.

November's poor showing among Latinos has led to a soul-searching for the GOP, which hopes to make gains among the growing voting bloc in coming years.

In the past, conservative Republicans have characterized any path to citizenship for illegal immigrants as "amnesty." Even McCain sharpened his rhetoric on the issue when he sought re-election in 2010, emphasizing border security and calling on the federal government to "complete the dang fence" separating the United States from Mexico, the largest single source of illegal immigration.

Obama, meanwhile, stepped up the use of National Guard troops to patrol the southern border, and deportations grew to nearly 400,000 in 2011. And the 2007-2009 recession in the United States helped cut sharply into the number of migrants in recent years, the Pew Hispanic Center reported in 2012.

Rep. David Schweikert, R-Arizona, told ABC that while the United States needs to draw talented immigrants, "if you go to the southern part of my state, I have people who actually live in fear for what's going through literally the back of their house, their property."

"The devil's ultimately in the details," Schweikert said. "What will happen to the populations today, but what will happen to our border security for those of us who are border states? What will happen in the visa system? Will we actually have a visa system that works, that tracks those who've overstayed, and then what does the future look like?"

But in a CNN-ORC International poll in mid-January, 53% of Americans said the main focus of U.S. policy on illegal immigration should be finding a way to allow undocumented residents to get legal status. By comparison, 43% said the priority should be border security and deportations.

"I'm cautiously optimistic," Menendez said. "I see the right spirit. I see things that were once off the table for agreement and discussion being on the table with a serious pathway forward."